Let A = {x ∈ R| cos x ∈ Z}, B = {x ∈ R| sin x ∈ Z}. Is A ⊆ B? Is B ⊆ A? Is either A or B a proper subset of the where other?

Answer 1

I'd say they are the same size.

All multiples of pi/2 are in both sets.

Then, since cosx = sin(pi/2 - x), wherever x is in A, pi/2-x is in B.